A new estimate for the number of solutions of a system of polynomial equations
Résumé
A theorem of Kuˇsnirenko and Bernˇstein shows that the number of isolated roots in the torus of a system of polynomials is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and that this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions.